Journal of Dynamics and Differential Equations

, Volume 23, Issue 1, pp 213–223

Local Fixed Point Indices of Iterations of Planar Maps


    • Faculty of Applied Physics and MathematicsGdansk University of Technology
  • Piotr Nowak-Przygodzki
    • Faculty of Applied Physics and MathematicsGdansk University of Technology
  • Francisco R. Ruiz del Portal
    • Departamento de Geometría y Topología Facultad de CC.MatemáticasUniversidad Complutense de Madrid
Open AccessArticle

DOI: 10.1007/s10884-011-9204-7

Cite this article as:
Graff, G., Nowak-Przygodzki, P. & Ruiz del Portal, F.R. J Dyn Diff Equat (2011) 23: 213. doi:10.1007/s10884-011-9204-7


Let \({f: U\rightarrow {\mathbb R}^2}\) be a continuous map, where U is an open subset of \({{\mathbb R}^2}\). We consider a fixed point p of f which is neither a sink nor a source and such that {p} is an isolated invariant set. Under these assumption we prove, using Conley index methods and Nielsen theory, that the sequence of fixed point indices of iterations \({\{{\rm ind}(f^n,p)\}_{n=1}^\infty}\) is periodic, bounded from above by 1, and has infinitely many non-positive terms, which is a generalization of Le Calvez and Yoccoz theorem (Annals of Math., 146, 241–293 (1997)) onto the class of non-injective maps. We apply our result to study the dynamics of continuous maps on 2-dimensional sphere.


Fixed point index Conley index Nielsen number Periodic points Iterations

Mathematics Subject Classification (2000)

Primary 37C25 Secondary 37E30 37B30

Copyright information

© Springer Science+Business Media, LLC 2011